Does a real constant affect asymptotic equivalence?

Question

The Meissel_Mertens constant $M = 0.26149...$ is defined as follows: $$ M := \lim_{n \to \infty} \left(\sum_{{p \in \mathbb{P}} \atop {p \leq n}} \frac{1}{p} - \ln(\ln n) \right)$$ Can we determine from that definition that $$\sum_{{p \in \mathbb{P}} \atop {p \leq n}} \frac{1}{p}\sim\ln(\ln n)?$$